1. Reference problem

1.1. Geometry

We represent a square with side \(l\mathrm{=}\mathrm{1m}\).

1.2. Illustration 1: Geometry of the test case

1.3. Material properties

Material:

Elasticity parameters:

Young’s modulus \({E}_{1}={20.10}^{9}\mathrm{MPa}\), Poisson’s ratio \({\nu }_{1}=\mathrm{0,25}\)

Parameters of law ENDO_HETEROGENE:

Elastic limit \({\sigma }_{y}\mathrm{=}5\mathit{MPa}\)

Weibull module \(m=6\)

Tenacity \({K}_{c}=1{\mathrm{MPa.m}}^{1/2}\)

Thickness of sample \(\mathit{ep}\mathrm{=}\mathrm{1m}\)

Seed \(\mathrm{GR}=121\)

Parameter of the non-local model:

Characteristic length \({l}_{c}=\mathrm{0,2}m\)

1.4. Boundary conditions and loading

The lower edge is blocked in movement in the vertical direction, the lateral edges are subject to a field of movement that varies linearly in height. Either:

At the bottom:

\({u}_{y}(x,y=0)=0\)

To the right

\({u}_{y}(x=l,y)={\mathrm{c.u}}_{d}(1-y/l)\)

with \({u}_{d}=\mathrm{0,0001}\)

To the right

\({u}_{y}(x=\mathrm{0,}y)=-{u}_{y}(x=l,y)\)

\(c\) corresponds to a loading ramp varying between 0 and 1 over the simulation time (1 s).

Figure 2: Boundary condition diagrams

1.5. Benchmark solution

There is no reference solution here and the test is of a non-regression type. It uses the coupling law ENDO_HETEROGENE, which is itself based on regularized constraints.

The result is therefore purely qualitative. The aim here is to observe the onset of a crack following lateral loading.